# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

93,656 questions
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### Is this set “not closed”? [duplicate]

Is it correct to say that this set $E=(0,1]$ where $E\subseteq R$ (Where $R$ is the set of real numbers) is not closed?
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### Probability of singletons in with an uncountable sample space

Let us assume that we have a sample space $\Omega=[0,1]$. Why is it not possible to have all the singletons $\{x\}\in\Omega$ with non-zero probability measure. I searched for an answer on this forum ...
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### If $f:U\subseteq\mathbb{R}^2\to\mathbb{R}$, $\partial f/\partial x$, $\partial f/\partial y$ exist and are continuous, then $f$ is differentiable [on hold]

Prove: If $f: U\subseteq \mathbb{R}^2\to\mathbb{R}$ and $\partial f/\partial x$, $\partial f/\partial y$ exist and are continuous on $U$, then $f$ is differentiable on $U$
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### Subset notation in Rudin's “Principles of Mathematical Analysis”

I've just began reading Walter Rudin's "Principles of Mathematical Analysis". Early on in the text is introduced subset notation, with which I am familiar. My problem is that Rudin doesn't seem to ...
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### Let : $P(x)=x^{3}+ax^{2}+bx+c$ where $(a,b,c)\in Z^3$

Question : If : $P(x)=x^3+ax^2+bx+c$ where $(a,b,c)\in Z^3$ And $m,n,k$ root of $P(x)$ such that : $m.n=k$ Then show that : $2P(-1)$ multiple of $P(1)+P(-1)-2[1+P(0)]$ My try : We known ...
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### Convert a real range to an integer range.

Let's say that we have a set of integers in the range $[1, 4]$. Now, I have a function that will calculate a distance between two vectors, and this function returns a real number in the range $[0, 1]$....
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### Convergence of indicator functions series

Say $E$ is a measurable set, and $\{E_k\}$ is a series of measurable sets defined by $$E_k \subset E, m(E \setminus E_k) < \frac{1}{k}, k = 1, 2, 3, ...$$ Do their corresponding indicator ...
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### A question about Abel Discriminance in series

I just read the proof of Abel discriminance in the series chapter. It says If ∑bn converges， and {an} is monotone with ｜an｜≤K， then series ∑anbn converges. in my ...
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### Calculate the sum.

There is given positive decreasing sequence of real numbers $a_{1},a_{2},a_{3},...$ Calculate the sum in nice form: $$SUM=\sum_{i=1}^{N}a_{i}-2\sum_{1\le i<j\le N}a_{i}a_{j}$$ If it is impossible ...
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### A ball contained in the image of a diffeomorphism

I am trying to prove the following. If $\varphi: U \subset \mathbb{R}^n \to V \subset \mathbb{R}^n$ is a diffeomorphism such that for $x\in B(0,r) \subset U$, $$|\varphi(x)-x| < \epsilon|x|,$$ ...
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### Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$

Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$ So in order to show injectivity $f(a)=f(b) \Rightarrow a=b$ so $\frac{a}{1-|a|}=\frac{b}{1-|b|}$. But how do I prove that? ...
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### Check if a set is open or close or neither

I've got the set G = {$x\in R^2: |x| \le 2, |y-1| \gt 1$} I need to determine whether this set is open or closed, I've drawn the set and I think it is neither, but I'm not sure how to proof it. ...
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### limit involving harmonic function

Let $u$ harmonic function in $\mathbb{R}^3 -\{0\}$. I know that $$\lim_{x\to0} \sqrt{|x|} \cdot u(x)=k< \infty$$ I'm trying to show that $k=0$. I tried by contradiction, but I failed and I'm ...
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### Is $p\neq 0$, then there are $x_+, x_- \in \mathbb{C}$ with $x_\pm ^2=p$.

Is $p\neq 0$, then there are $x_+, x_- \in \mathbb{C}$ with $x_\pm ^2=p$. My proof: The proof for $p>0$ is trivial, since $\mathbb{R}\subset \mathbb{C}$ and it's true $\forall p\in \mathbb{R}$. ...
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